Projection operators and states in the tensor product of quaternion hilbert modules

Following the construction of tensor product spaces of quaternion Hilbert modules in our previous paper, we define the analogue of a ray (in a complex quantum mechanics) and the corresponding projection operator, and through these the notion of a state and density operators. We find that there is a one-to-one correspondence between a state and an equivalence class of vectors from the tensor product space, which gives us another method to define the gauge transformations.On sabbatical leave from the School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel. Work supported in part by a fellowship from the Ambrose Monell Foundation.     

Extending Scope of Robust Optimization: Comprehensive Robust Counterparts of Uncertain Problems

In this paper, we propose a new methodology for handling optimization problems with uncertain data. With the usual Robust Optimization paradigm, one looks for the decisions ensuring a required performance for all realizations of the data from a given bounded uncertainty set, whereas with the proposed approach, we require also a controlled deterioration in performance when the data is outside the uncertainty set. The extension of Robust Optimization methodology developed in this paper opens up new possibilities to solve efficiently multi-stage finite-horizon uncertain optimization problems, in particular, to analyze and to synthesize linear controllers for discrete time dynamical systems. Research was supported by the Binational Science Foundation grant #2002038     

Robust optimization – methodology and applications

Robust Optimization (RO) is a modeling methodology, combined with computational tools, to process optimization problems in which the data are uncertain and is only known to belong to some uncertainty set. The paper surveys the main results of RO as applied to uncertain linear, conic quadratic and semidefinite programming. For these cases, computationally tractable robust counterparts of uncertain problems are explicitly obtained, or good approximations of these counterparts are proposed, making RO a useful tool for real-world applications. We discuss some of these applications, specifically: antenna design, truss topology design and stability analysis/synthesis in uncertain dynamic systems. We also describe a case study of 90 LPs from the NETLIB collection. The study reveals that the feasibility properties of the usual solutions of real world LPs can be severely affected by small perturbations of the data and that the RO methodology can be successfully used to overcome this phenomenon. Received: May 24, 2000 / Accepted: September 12, 2001?Published online February 14, 2002     

Robust solutions of Linear Programming problems contaminated with uncertain data

Optimal solutions of Linear Programming problems may become severely infeasible if the nominal data is slightly perturbed. We demonstrate this phenomenon by studying 90 LPs from the well-known NETLIB collection. We then apply the Robust Optimization methodology (Ben-Tal and Nemirovski [1–3]; El Ghaoui et al. [5, 6]) to produce “robust” solutions of the above LPs which are in a sense immuned against uncertainty. Surprisingly, for the NETLIB problems these robust solutions nearly lose nothing in optimality. Received: July 1999 / Accepted: May 2000?Published online July 20, 2000     

Robust multi-echelon multi-period inventory control

We consider the problem of minimizing the overall cost of a supply chain, over a possible long horizon, under demand uncertainly which is known only crudely. Under such circumstances, the method of choice is Robust Optimization, in particular the Affinely Adjustable Robust Counterpart (AARC) method which leads to tractable deterministic optimization problems. The latter is due to a recent re-parametrization technique for discrete time linear control systems. In this paper we model, analyze and test an extension of the AARC method known as the Globalized Robust Counterpart (GRC) in order to control inventories in serial supply chains. A simulation study demonstrates the merit of the methods employed here, in particular, it shows that a good control law that minimizes cost achieves simultaneously good control of the bullwhip effect.     

Achievement and assessment of direct electron transfer of glucose oxidase in electrochemical biosensing using carbon nanotubes,graphene, and their nanocomposites

Carbon nanotubes, graphenes, and their hybridized composites with nanoparticles have been attempted to establish a direct electrical communication between the recognition biomolecule and its underlying electrode surface. This review (with 133 refs.) focuses on advances, strategies and technical challenges in the development of reagentless electrochemical biosensors for glucose with enhanced detection sensitivity, selectivity, and simplicity. Specifically, the review commences with a discussion of the relevance of direct electron transfer (DET) in biosensing together with the fundamental of electro-enzymology and kinetics. General aspects of glucose oxidase (GOx), the most popular enzyme with a flavin cofactor, are discussed in view of its historical and important role in the development of electrical biosensors for blood glucose. The next section assesses DET of GOx based on the Marcus theory and the Laviron formalism. The reorganizational energy of the Marcus model and the overpotential play an important role in reaction kinetics and affect the rate of electron transfer significantly. The presence of nanomaterials, particularly for graphene oxide, decreases the electron transfer distance between the enzyme redox center and the underlying electrode surface well beyond 15 Å. The improper Marcus-Hush-Chidsey integral is now simplified to estimate the rate of electron transfer with very good accuracy. Critiques, technical challenges, and future possibilities of glucose electrodes with respect to DET are also presented and discussed.

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Graphical abstract This review (with 133 refs.) focuses on advances, strategies and technical challenges in the development of reagentless electrochemical biosensors for glucose with enhanced detection sensitivity, selectivity, and simplicity.

     

Electro-optic beam manipulation through photorefractive needles

We demonstrate electro-optic spatial two-dimensional mode switching in a bulk sample of potassium lithium tantalate niobate. Spatial confinement, mode coupling, and electro-optic functionality are mediated by two photorefractive needle solitons of opposite electroholographic charges embedded together in their anisotropic lobular structure.     

Microwave-assisted polyol synthesis of CuInTe2 and CuInSe2 nanoparticles

Nanoparticles of the CuInTe(2) (CIT) and CuInSe(2) (CIS) alloys have been prepared using the microwave-assisted polyol method. In this simple and quick reaction the polyol is both the solvent and the reducing agent. XRD studies show that nanoparticles of CIT and CIS are formed in the body-centered tetragonal structure and their average diameters are approximately 94 and approximately 83 nm, respectively. Electron microscopy studies show that these formed particles are poorly aggregated with a mean diameter of 100 and 85 nm, respectively. The products have been characterized by different analytical techniques, and the electronic properties have been measured using photoacoustic spectroscopy (PAS).     

Tensor product of quaternion hilbert modules

One of the main problems in the theory of quaternion quantum mechanics has been the construction of a tensor product of quaternion Hilbert modules. A solution to this problem is given by studying the tensor product of quaternion algebras (over the reals) and some of its quotient modules. Real, complex, and (covariant) quaternion scalar products are found in the tensor product spaces. Annihilationcreation operators are constructed, corresponding to the second quantization of the quaternion quantum theory with Bose-Einstein or Fermi-Dirac statistics. The gauge transformations of a tensor product vector and the gauge fields are studied.On Sabbatical leave from the School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel. Work supported in part by a fellowship from the Ambrose Monell Foundation.     

Global minimization by reducing the duality gap

We derive a general principle demonstrating that by partitioning the feasible set, the duality gap, existing between a nonconvex program and its lagrangian dual, can be reduced, and in important special cases, even eliminated. The principle can be implemented in a Branch and Bound algorithm which computes an approximate global solution and a corresponding lower bound on the global optimal value. The algorithm involves decomposition and a nonsmooth local search. Numerical results for applying the algorithm to the pooling problem in oil refineries are given.Research supported by Shell Laboratorium, Amsterdam, and GIF—The German—Israel Foundation for Scientific Research and Development.